MUCH OF THE WORLD S MUSIC involves

Production and Synchronization of Uneven Rhythms at Fast Tempi 61 PRODUCTION AND SYNCHRONIZATION OF UNEVEN RHYTHMS AT FAST TEMPI BRUNO H. REPP Haskins Laboratories, New Haven, Connecticut JUSTIN LONDON Carleton College, Northfield, Minnesota PETER E. KELLER Max Planck Institute for Human Cognitive and Brain Sciences, Munich, Germany THIS STUDY EXAMINED PRODUCTION and synchronization of eight uneven rhythms (set A: 2 3, 3 2; set B: 2 2 3, 2 3 2, 3 2 2; set C: 2 3 3, 3 2 3, 3 3 2) at rates that made it difficult to subdivide the component intervals into elementary metrical pulses. The main questions were how interval ratios would change as a function of tempo within this range, and whether metrical downbeat location (which distinguishes the rhythms within each set) would be reflected in any of the dependent measures. Musically trained participants tapped each rhythm cyclically in synchrony with an auditory template and then continued to tap in three ways: self-paced, paced by a sequence of downbeats, or paced by a rapid stream of isochronous subdivisions. Each task was carried out at eight tempi. The third task assessed the temporal limit of subdivision for these rhythms (about 6 Hz), which was exceeded by most tempi. Results showed that the mean long short (3:2) interval ratio was already larger than 1.5 at the slowest tempo in rhythm sets A and B, and increased with tempo in sets B and C, but did not approach 2. Uneven rhythms thus can be produced without mental subdivision, but only with substantial enhancement of the contrast between long and short intervals. Metrical downbeat location had no reliable effect on interval ratios but was reflected in more forceful downbeat taps and in different alignments of taps with a pacing sequence. In general, effects of temporal grouping (between rhythm sets) outweighed those of metrical interpretation (within rhythm sets). Received December 2, 2003, accepted January 8, 2005 Uneven Rhythms MUCH OF THE WORLD S MUSIC involves uneven rhythms, from African drum patterns to Norwegian Hardanger fiddle tunes. By uneven we mean a special relationship between long and short elements in a rhythmic pattern. Consider two cyclically repeated rhythmic patterns, one composed of a quarter note followed by two eighth notes, and the other composed of a dotted eighth note followed by two eighth notes. In both patterns, the nominal durations of the inter-onset intervals (IOIs) are categorically different, but in the first pattern the long duration is the sum of the two short ones, and so this rhythm can be (and most likely will be) construed in the context of a meter composed of isochronous beat periods, such as 2/4 time. The second pattern cannot fit into a similarly isochronous meter because its nominal IOI durations exhibit a 3:2:2 relationship. Even though the underpinning subdivisions (16th notes) are isochronous, and the highest level of periodicity is isochronous (the pattern repeats every seven 8th notes), the middle level is not. Its metric construal hearing the pattern as a series of uneven beats requires a complex meter such as 7/8 time (London, 1995). Uneven rhythms, while well-known in the musical (and especially ethnomusicological) literature, have received relatively little empirical study regarding their production and/or perception. One recent exception is Magill and Pressing s (1997) investigation of rhythms produced by an African master drummer, which showed by means of statistical analyses that the rhythms had been generated as an additive pattern of uneven beats (considered idiomatic in West African music) rather than a series of intervals superimposed on regular subdivisions (as a Western musician might suppose). Unlike that interdisciplinary study which concerned music of a different culture, used a single expert performer, and employed modeling of interval covariance structure the present research is more in the tradition of rhythm production and synchronization studies in the mainstream psychological literature. Specifically, we are concerned with the challenges that uneven rhythms pose to individuals who are trained in Music Perception VOLUME 23, ISSUE 1, PP. 61-78, ISSN 0730-7829, ELECTRONIC ISSN 1533-8312 2005 BY THE REGENTS OF THE UNIVERSITY OF CALIFORNIA. ALL RIGHTS RESERVED. PLEASE DIRECT ALL REQUESTS FOR PERMISSION TO PHOTOCOPY OR REPRODUCE ARTICLE CONTENT THROUGH THE UNIVERSITY OF CALIFORNIA PRESS S RIGHTS AND PERMISSIONS WEBSITE AT WWW.UCPRESS.EDU/JOURNALS/RIGHTS.HTM

62 B. H. Repp, J. London and P. E. Keller the Western classical musical tradition, in which such rhythms are relatively rare. Pioneering research on the production of uneven rhythmic patterns was carried out by Fraisse (1956). He requested his participants to tap spontaneously generated rhythms without repetition, thereby (perhaps deliberately) discouraging the development of metrical frameworks. Fraisse observed that long and short intervals within these rhythms tended toward a 2:1 ratio, although actually a wide range of ratios occurred, from about 1.5 to 3. The average ratio tended to be maximal at a moderate tempo and decreased as the tempo was either increased or decreased. In a later replication, Essens and Povel (1985, Exp. 1) found a much narrower range of ratios and a closer match to 2:1 for spontaneously generated rhythms, probably because the rhythms were repeated cyclically and thus induced a metrical structure that constrained the interval ratio. Tempo was not varied systematically; the tempo variations that occurred spontaneously did not seem to have any effect on the interval ratio. Povel (1981) demonstrated that cyclically repeated rhythms are easier to reproduce when their component intervals have simple rather than complex ratios. Participants reproduced a 2:1 interval ratio quite accurately but were unable to reproduce a 3:2 ratio even when the tempo was slow enough to allow the counting of subdivisions (600 400 ms). The reproduced ratio approached 2:1, which implies a sharpening of the contrast between the long and short intervals. Less pronounced sharpening was observed in sequences composed of three (2:3:3), five (2:2:2:3:3), or seven intervals (2:2:3:2:3:2:3; Essens & Povel, 1985, Exp. 2), but even musically trained participants were unable to reproduce these rhythms accurately. Povel attributed this to an inability to encode the rhythms as simple hierarchical structures. Other researchers who have obtained results consistent with these findings include Sternberg, Knoll, and Zukofsky (1982), Summers, Bell, and Burns (1989), and Semjen and Ivry (2001). In a recent study of the effect of tempo on rhythm production (Repp, Windsor, & Desain, 2002), skilled pianists played melodies notated in various two- and three-interval rhythms at four different tempi ranging from moderate to very fast. The two-interval rhythms included the uneven rhythms 2 3 and 3 2, notated in 5/8 meter. Results showed that the 2 3 rhythm was played at a ratio close to 1:2 at all tempi, whereas the 3 2 rhythm was played more accurately on average. This suggests a possible effect of the order of the intervals within the rhythmic cycle, which amounts to an effect of metrical structure (viz., of downbeat location). 1 Production of 1 2 and 2 1 rhythms, notated in 3/8 meter, was generally accurate. While tempo had little effect on the production of two-interval rhythms, it had a strong effect on the production of three-interval rhythms consisting of all possible permutations of intervals in 1:2:3 ratios, notated in either 3/4 or 6/8 meter. In those rhythms which can be strongly syncopated but are not uneven by our definition because they can be notated in standard meters the two longer intervals became increasingly similar in duration as the tempo increased, whereas the short interval remained proportionally stable. This progressive reduction of the 3:2 ratio in even three-interval rhythms contrasts with the seemingly tempo-invariant sharpening of the same ratio in uneven two-interval rhythms, and it led us to wonder whether and how tempo would affect the production of uneven threeinterval rhythms. We should note that, in the context of real music performance (as opposed to simple rhythm production), temporal intervals are rarely performed with the exact integer ratios suggested by notation, and this contributes to the special feeling and character of a rhythm. For example, Gabrielsson, Bengtsson, and Gabrielsson (1983) found that 2:1 and 3:1 ratios were generally reduced by musicians playing simple melodies on various instruments. Friberg and Sundström (2002) showed that the swing ratio of jazz drummers (nominally a 1:1 ratio) varied from 1:1 to 3:1 as a function of tempo. When two short notes are followed by a longer note, their nominal 1:1 ratio is commonly rendered as a shortlong pattern, and this distortion occurs even when participants are instructed to play with mechanical accuracy (Drake & Palmer, 1993; Repp, 1999). Thus, while some deviations from nominal ratios may be intentional and serve expressive purposes, other deviations seem to be obligatory and hard to avoid, even in tasks that merely require exact production of simple rhythms. The present study concerns such obligatory distortions of interval ratios when exact performance is intended, specifically of the 3:2 ratio in the context of uneven rhythms, where that ratio is of paramount importance. The materials included not only the 2 3 and 3 2 rhythms already studied by Repp et al. (2002), referred to as set A, but also uneven rhythms consisting of three intervals per measure, sets B (2 2 3, 2 3 2, 3 2 2) and C (2 3 3, 3 2 3, 3 3 2). The task was finger tapping rather than performance on a musical instrument. 1 Two pianists, however, produced the 3 2 rhythm as 1 1, which contributed to the overall difference between 2 3 and 3 2.

Production and Synchronization of Uneven Rhythms at Fast Tempi 63 Our study had three main purposes, which are introduced below. One was to determine the temporal limit of metrical subdivision for uneven rhythms. Another purpose was to study how tempo and metrical interpretation influence uneven rhythm production. The third purpose was to study how uneven rhythms are aligned with pacing sequences in a synchronization task. Temporal Limits of Metrical Subdivision The importance of mental subdivision ( counting ) is often emphasized in texts on Western music performance, especially in connection with unfamiliar or complex rhythms. For example, Weisberg (1993) writes that it is always necessary to keep the speed of the common unit (in the case of 5/8, the eighth note) in mind (p. 5) and further that musicians must develop an inner way of counting, one that wastes as little energy as possible. The counting must be silent and internal (p. 6). Therefore, when an uneven rhythm such as 3 2 is to be produced with maximum accuracy, we would certainly expect musicians to employ mental subdivision (as long as the tempo permits it), even if they are not specifically instructed to do so. Somewhat surprisingly, the findings of Povel (1981) and Essens and Povel (1985), reviewed above, suggest that production of the 3:2 ratio is inaccurate (tending toward 2:1) even when counting seems possible and even when the participants are musically trained. We wondered whether we could replicate that finding. We also wanted to determine the temporal limit of subdivision for uneven rhythms. London (2002, 2004) reviewed both the music-theoretic and psychological literature on rhythm perception and performance and concluded that the shortest IOI that can be perceived or performed as an element of a rhythmic pattern is around 100 ms. This hypothesis has found support in a recent study of sensorimotor synchronization (finger tapping) with isochronous auditory sequences (Repp, 2003), as well as in the study of jazz performance by Friberg and Sundström (2002), cited above. Thus, one might hypothesize that the shortest IOIs enabling a 3:2 ratio to be produced with mental subdivision would be multiples of 100 ms (i.e., 300 and 200 ms). However, the cognitive demands of alternating between counting 3 and 2 may well lead to an elevated subdivision limit for uneven rhythms. To determine that limit, we required participants to synchronize uneven rhythms with an isochronous sequence of rapid subdivisions. We assumed that the sequence rate at which synchronization with explicit subdivisions becomes impossible is also an estimate of the maximal rate at which mental subdivision can be accomplished. Rhythm Production We assessed rhythm production primarily in terms of interval ratios, with additional information provided by measures of interval variability and tapping force. As already mentioned, we wondered whether musically trained participants would distort the 3:2 ratio already at the slowest tempo, where we presumed subdivision would still be possible. Furthermore, we wanted to determine whether the produced ratio changes as the tempo is increased beyond the temporal limit of subdivision. We also examined whether rhythm production is more accurate when participants synchronize with a precise rhythm template than when they produce the rhythm freely (i.e., in a self-paced manner). In addition, we were interested in differences between and within the three rhythm sets. These differences can be characterized in terms of temporal grouping (or interval structure) and metrical interpretation, respectively. Table 1 illustrates these distinctions. Because uneven rhythms contain unequal IOIs, they consist of temporally grouped events when they are repeated cyclically. Short IOIs function as within-group intervals; longer IOIs, as between-group intervals. It can be seen in Table 1 (where timing is rendered as spacing) that the rhythms in sets A and B form temporal groups of two and three events, respectively. In set C, the temporal grouping is ambiguous because a single event alternates with a two-event group. Assuming that a single event cannot form a group by itself, it could be regarded either as the initial or final element of a three-element group. For purposes of analysis, we assumed the second of these two options, based on the reasoning that it TABLE 1. The three sets of rhythms. Vertical bars symbolize taps or tones; dots are silent subdivisions. Three cycles of each rhythm are shown, followed by an additional downbeat. Set A 2 3 q q.......... 3 2 q. q......... Set B 2 2 3 q q q............. 2 3 2 q q. q............ 3 2 2 q. q q............ Set C 2 3 3 q q. q................ 3 3 2 q. q q................ 3 3 2 q. q. q............... *q=quarter-note

64 B. H. Repp, J. London and P. E. Keller seems more natural for the long within-group IOI to follow the short IOI. (For comments on cognitive grouping, as distinct from temporal grouping, see the General Discussion.) The rhythms within each set differ in the location of the metrical downbeat (which, at fast tempi, may simply be the beat or tactus). The downbeat is the initial element of the measure or rhythm cycle, as defined in instructions and shown in quasi-musical notation in Table 1. In the schematic sequences of Table 1, downbeats are vertically aligned within each rhythm set. In set A, the downbeat can fall on either the initial (2 3) or the final (3 2) element of the temporal group. In sets B and C, the downbeat can be group-initial (2 2 3, 2 3 3), group-medial (2 3 2, 3 2 3), or group-final (3 2 2, 3 3 2). The placement of the downbeat is a cognitive act, reflecting a metrical interpretation of a constant temporal grouping structure, and we wondered whether this cognitive act would become manifest in interval ratios or other aspects of rhythm production. One might hypothesize, for example, that participants will emphasize the metrical downbeat by lengthening the subsequent inter-tap interval or by making a more forceful tap. Although temporal grouping and metrical interpretation were varied orthogonally in our materials, they are not completely independent. It is known that, in a group of two identical tones, the second tone tends to be heard as accented (Povel & Okkerman, 1981), whereas in a group of three tones, the first and third tones are so perceived (Povel & Essens, 1985). This grouping accent is a major factor in beat induction from a rhythmic sequence (Lerdahl & Jackendoff, 1983; Povel & Essens, 1985). If the location of the metrical downbeat is manipulated within a fixed grouping structure by means of instructions, then in some rhythms the downbeat coincides with a grouping accent and in others it does not. The former rhythms therefore may be easier to produce than the latter. Thus, 3 2 should be easier than 2 3, 2 2 3 and 3 2 2 should be easier than 2 3 2, and 2 3 3 and 3 2 3 may be easier than 3 3 2 (if our grouping assumption is correct). Relative difficulty of production was expected to be reflected mainly in interval variability. Regardless of metrical interpretation, temporal grouping structure was expected to have strong effects on interval variability and the relative force of taps. It is well-known that long intervals are more variable than short ones (Peters, 1989), even in mixed sequences (Repp, 1997), and taps preceded by a long inter-tap interval tend to be more forceful than those preceded by a short interval because there is more time to lift the finger (Repp & Saltzman, 2002). Synchronization We examined synchronization with an exact auditory rhythm template and with an isochronous sequence of downbeats. A third condition, synchronization with a rapid isochronous sequence of subdivisions, served the special purpose of determining the temporal limit of subdivision, as described earlier. The dependent variables were the tap-tone asynchronies and their variability. In synchronization with a rhythm template, possible effects of metrical interpretation were of primary interest. Our hypothesis was that the mean asynchrony associated with a particular temporal group position might be smaller and/or less variable when the metrical downbeat falls in that position. A comparison of downbeat asynchronies between the rhythm template and downbeat synchronization conditions addressed the question of whether downbeat asynchronies are sensitive to the presence or absence of metrically weak events in the pacing sequence. Method Participants The 8 participants (4 women, 4 men) included 7 paid volunteers and one of the authors (B.R.). 2 All were musically trained and were also regular participants in rhythmic finger-tapping experiments. Four were professional-level classical musicians (two violists, one clarinetist, and one percussionist) and the other four were advanced amateur pianists, three with classical training and one (B.S.) who played jazz piano as well as African drums. Two of the amateurs (B.S., B.R.) were 57 years old; other participants ranged in age from 19 to 30. Materials The eight rhythms investigated constitute three sets (see Table 1): (A) two-interval rhythms (2 3, 3 2); (B) three-interval rhythms containing two short intervals (2 2 3, 2 3 2, 3 2 2); and (C) three-interval 2 B.R. had previous experience with the tasks from a pilot run of the experiment, but 8 months elapsed before he ran himself again in the final version. One additional participant s data could not be used because he often tapped too lightly at fast tempi, so that many taps were not registered.

Production and Synchronization of Uneven Rhythms at Fast Tempi 65 TABLE 2. Tasks in the three sessions of the experiment. The diagrams show the transition from the synchronization to the continuation phase for the 2 3 rhythm. Session I Synchronization Continuation Tones......... Taps.................. Session II Synchronization Continuation Tones..................... Taps.................. Session III Synchronization Continuation Tones......... Taps.................. rhythms containing two long intervals (3 3 2, 3 2 3, 2 3 3). In the analysis and description of results, the two nominally equal intervals in the three-interval rhythms will be distinguished according to whether they immediately preceded or followed the third interval during cyclic repetition: 2b (before 3) versus 2a (after 3) in set B, and 3b (before 2) versus 3a (after 2) in set C. For example, 2 2 3 2a 2b 3, and 2 3 3 2 3a 3b. The musical time signatures of the three sets, if they were notated, would have to be 5/8, 7/8, and 8/8, respectively. 3 Each rhythm was produced at increasing tempi in three synchronization-continuation tapping tasks that differed in the nature of the continuation task, as illustrated in Table 2. During the synchronization phase of each task, participants were required to tap in synchrony with a computer-generated auditory sequence that instantiated the rhythm with mathematical accuracy (i.e., a rhythm template). Subsequently, they continued to tap the rhythm in a self-paced manner (Session I) or were paced by an isochronous series of downbeats (Session II) or by a rapid isochronous series of subdivisions (Session III). Although the continuation tasks of Sessions II and III were synchronization tasks as well, we call them continuation tasks whenever they need to be distinguished from the initial synchronization with a rhythm template that was common to Sessions I, II, and III. 3 The rhythms in 8/8 meter could be construed as syncopated rhythms within an even 4/4 metrical framework. We found it unlikely, however, that this interpretation would be adopted by participants in our experiment because it is cognitively much more demanding than an uneven meter. All pacing sequences consisted of identical highpitched digital piano tones (E7, about 2640 Hz), which were produced on a Roland RD-250 digital piano under control of MAX 3.0 software running on a Macintosh Quadra 660AV computer. 4 The tones had abrupt onsets and decayed freely thereafter in a roughly exponential fashion. Participants listened over Sennheiser HD540 II headphones at a comfortable intensity. In Session I, each trial consisted of 15 cycles of a particular rhythm template plus one extra tone (downbeat) at the end, followed by a silent continuation period equivalent to the duration of 29 cycles; a single tone signaled the end of that period. In Session II, each trial began likewise with 15 cycles of a rhythm template and then continued with 30 tones marking cycle beginnings (downbeats). In Session III, each trial began with 7 cycles of a rhythm template and then continued with rapid isochronous subdivision beats for the equivalent of 20 pattern cycles. Because the task of Session III served a different purpose (the determination of the subdivision limit), the abbreviated synchronization condition only served to induce the rhythm; the data from that condition were not analyzed. For each rhythm in each session, there were eight successive trials differing in tempo. Tempo was defined in terms of the duration of (implicit or explicit) subdivisions, referred to here as metrical grid spacing (MGS), which decreased from 170 ms in the first trial to 100 ms in the eighth trial, in steps of 10 ms. For example, the interval durations for the 2 3 2 pattern with MGS 140 ms were 280, 420, and 280 ms, and the cycle duration was 980 ms. The cycle durations of the rhythms in the three sets were 5*MGS, 7*MGS, and 8*MGS, respectively. The range of MGS values was selected by author B.R. on the basis of his impression during a pilot run that mental subdivision was possible at 170 ms but clearly impossible at 100 ms. Procedure The three sessions were typically one week apart and lasted a good hour (I, II) or less than an hour (III). 4 Due to a peculiarity of this setup, the tempo of the output was about 2.4% faster than specified in the MIDI instructions, as had been determined in earlier acoustic waveform measurements. The participants finger taps were registered at a correspondingly slower rate. Throughout this article, all millisecond values are reported as they appeared in the MAX environment. Apart from the constant scaling factor of.976, MAX was highly accurate (within 1 ms) in timing the sequences and registering the finger taps.

66 B. H. Repp, J. London and P. E. Keller Within each session, the three rhythm sets were performed in the order A, B, C. The order of the rhythms within each set varied across participants but was the same in each session for each participant. At the beginning of each session, the participant read written instructions containing illustrations like those in Tables 1 and 2, and he or she was given a few sample trials. Each rhythm to be produced was announced by the experimenter and was pointed to on the instruction sheet; participants had time to try out each rhythm informally before starting a series of trials. In Session I, each trial required synchronization of taps with an auditory rhythm template, starting with the third downbeat in the sequence (i.e., the fifth tone for rhythm set A, the seventh tone for sets B and C), and continuation of the rhythm after the sequence ended, without interruption and at the same tempo, until a single tone sounded (self-paced continuation). In Session II, the task was similar, except that during continuation of the rhythm each downbeat tap had to be synchronized with a tone (downbeat-paced continuation). The downbeat tap was defined as the one that started each cycle of a rhythm (see Table 1). In Session III, after the abbreviated initial synchronization phase, each tap had to be synchronized with a subdivision tone (subdivisionpaced continuation). Participants were told that, if they could not synchronize with the appropriate subdivision tones, they should ignore them and simply tap the rhythm as regularly as possible until the subdivision sequence ended. Participants sat in front of a computer monitor that displayed the trial number. The first trial in a block of eight trials for a given rhythm was started by clicking a button with the mouse, and each subsequent trial was started by the participant pressing the space bar of the computer keyboard. If a participant started tapping at the wrong time or was unhappy with his or her performance on a trial, the trial was repeated, but such repetitions were infrequent. There were short breaks between blocks of trials. Participants tapped with the preferred (right) hand on a Roland SPD-6 electronic percussion pad, which they held on their lap. The sound output of the percussion pad was not used, but there was direct auditory feedback from the taps (a thud), in proportion to the tapping force. The two percussionists chose to tap forcefully from above using the middle finger, whereas other participants tapped more gently with the index finger and typically rested their palm and other fingers on the pad while tapping. The taps were registered by the percussion pad (set to manual sensitivity) and were transmitted as MIDI information to the computer. Data Analysis At fast tempi, it sometimes happened that some taps were too weak to be registered electronically. These occasional gaps in the data were corrected by realigning the registered taps when computing asynchronies and by deleting inter-tap intervals that spanned a gap. The gaps were generally not a problem for computation of means and standard deviations. In some instances, however, they were so frequent and/or the produced rhythm was so distorted that the trial was considered a failure to execute the required task. Such trials were deleted, as were rare trials that exhibited phase slips or continuous phase drift in the downbeat-paced continuation phase of Session II. (Trials exhibiting phase drift in Session III were not excluded; see below.) There were no categorical mistakes in rhythm production, nor were there any failures to synchronize with the rhythm template during the synchronization phases of Sessions I and II. The total number of trials excluded from analysis was 29 (1.9%), nearly all of them at the fastest tempi. For the analysis of rhythm production, mean intertap interval durations and standard deviations were computed across all intact cycles of each rhythm in each trial, separately for the synchronization and continuation conditions in Sessions I and II, and also for the continuation condition in Session III. Interval ratios were computed from the mean interval durations of each trial by dividing the duration of a long ( 3 ) interval by that of a short ( 2 ) interval, so that the nominal (expected) ratio was 1.5 in all cases. For the rhythms in set A, there was just one ratio (3/2); for each of the other two rhythm sets, there were two ratios (set B: 3/2a and 3/2b; set C: 3a/2 and 3b/2). The data from Sessions I and II were subjected to separate repeatedmeasures ANOVAs for each rhythm set, with the variables of ratio (2 levels, for sets B and C only), rhythm (2 levels in set A, 3 levels in sets B and C), session (2 levels), condition (2 levels: synchronization vs. continuation), and tempo (8 levels). Missing cells due to excluded trials were filled in by duplicating the value(s) of the most appropriate adjacent cells in the design. The continuation data from Session III were analyzed together with the continuation data from Sessions I and II in a second set of ANOVAs, whose results will be mentioned only if they are of special interest. For the analysis of synchronization with a rhythm template (Sessions I and II), asynchronies were computed by subtracting the pacing sequence tone onset times from the tap registration times. Thus, a negative asynchrony indicates that the tap preceded the tone.

Production and Synchronization of Uneven Rhythms at Fast Tempi 67 Mean asynchronies and standard deviations were computed across rhythm cycles and analyzed in separate repeated-measures ANOVAs for each rhythm set, with the variables of position (2 or 3 levels), rhythm (2 or 3 levels), session (2 levels), and tempo (8 levels). Position was defined with respect to the temporal grouping structure of a rhythm set and was independent of downbeat location, which was represented by the rhythm variable. The asynchronies of the downbeatpaced continuation condition in Session II were analyzed separately together with the downbeat asynchronies extracted from the synchronization condition of Session II. The asynchronies of the subdivisionpaced continuation tapping in Session III were not analyzed in detail. In addition to these temporal measurements, MIDI velocities (range: 0 127) were obtained as a rough measure of tapping force. One participant, the professional percussionist, tapped so strongly that the MIDI velocity was usually at its maximum; his data had to be excluded from analysis. Only the MIDI velocities from Session I were analyzed, in ANOVAs similar to those on asynchronies. In order not to clutter the text too much, ANOVA results are not reported exhaustively. Statistics are omitted for effects so large that their significance is obvious, as well as for duplicate results and for minor effects that are of little theoretical interest. Although the Greenhouse-Geisser or Huyhn-Feldt correction is recommended for repeated-measures effects with more than one degree of freedom, the computer program used did not provide either of these corrections. To compensate for this, multiple-degrees-of-freedom effects were considered significant only if p.01. Results Temporal Limits of Metrical Subdivision The purpose of the subdivision-paced continuation condition of Session III was to provide an estimate of the temporal limit of metrical subdivision for uneven rhythms, a limit that presumably also applies to mental subdivision when subdivisions are not physically present. Failures to synchronize were apparent as phase drift relative to the pacing sequence during part or all of the continuation tapping. A trial was considered successful if the standard deviation of the downbeat asynchronies, calculated relative to the theoretical downbeat in the pacing sequence, was less than 50 ms. Trials with larger variability were considered unsuccessful. This criterion was arbitrary, but it distinguished well between trials that exhibited phase drift (which is easy to detect by eye FIG 1. Percentage of successful trials in the subdivision-paced continuation condition of Session III as a function of metrical grid spacing. in a series of numerical asynchrony values) and those that did not. Borderline cases (e.g., where phase drift started near the end of a trial) were rare. Figure 1 shows the mean percentage of successful trials as a function of MGS duration. The range of tempi (MGS durations) had been chosen with the expectation that synchronization with a rapid stream of subdivisions would be possible at the slowest tempo, but participants found the task somewhat more difficult than expected. On average, synchronization was successful only about 60% of the time at the slowest tempo, and the percentage decreased rapidly as the tempo increased. If the 50% crossover point is taken as an estimate of the mean synchronization threshold, then the estimate is 163 ms, which is considerably higher than a previous estimate of 123 ms for tapping with every fourth tone of an isochronous sequence (Repp, 2003). This suggests that the temporal limit of subdivision increases with the complexity of the rhythm. Individual synchronization thresholds ranged from 143 ms (for the professional percussionist) to above 170 ms (for the participant with the least musical training). The large majority of unsuccessful trials exhibited a slowing down of tapping relative to the pacing sequence, resulting in increasingly large positive asynchronies. Only one participant had about equal numbers of unsuccessful trials showing slowing down and speeding up.

68 B. H. Repp, J. London and P. E. Keller The data also suggested some differences among rhythms, although these data were too sparse for statistical significance tests. Synchronization success was lower for 3 2 than for 2 3, lower for 2 3 2 than for 2 2 3 and 3 2 2, and lower for 3 2 3 than for 3 3 2 and 2 3 3. The differences among the threeinterval rhythms in sets B and C are in accord with the prediction that rhythms whose downbeat does not coincide with a grouping accent should be more difficult to execute. The difference in set A, however, is contrary to this prediction. In summary, the results of this part of our study suggest that, within the range of chosen tempi, subdivision of uneven rhythms into elementary metrical pulses was difficult to begin with and became impossible as the tempo increased. Except for the initial difficulty, which was greater than expected, this was the situation we had intended to create for our examination of rhythm production and synchronization. Rhythm Production INTERVAL RATIOS The three main questions regarding interval ratios were as follows: (a) What ratio do participants start out with at the slowest tempo? (b) How does the ratio change as the tempo is increased? (c) Does metrical downbeat location have any effect on the interval ratio? We address these questions for each rhythm set in turn. For the two-interval rhythms in set A, Repp et al. (2002) had found that the interval ratio was sharpened considerably (i.e., greater than 1.5) even at a slow tempo and did not change significantly as the tempo increased. This finding was basically replicated: The mean interval ratio for set A rhythms in Sessions I and II was 1.76, and the main effect of tempo was nonsignificant, as were all interactions involving tempo. However, closer inspection of the data revealed considerable individual differences, which are summarized in Table 3. The significance of individual changes in interval ratio was determined by computing the correlation between interval ratio and MGS duration across 2 (rhythms) 8 (tempi) 5 (conditions) 80 data points for each participant, where the conditions include synchronization in Sessions I and II and continuation in Sessions I III. For three participants, mean interval ratio increased significantly as MSG duration decreased (a negative correlation), whereas for three others the ratio decreased (a positive correlation); two showed no significant change. The main effect of rhythm (2 3 vs. 3 2) was nonsignificant in set A; thus, there was no overall effect TABLE 3. Mean interval ratios (intercepts of regression line relating interval ratio to MGS duration) for set A rhythms at the slowest (MGS 170 ms) and fastest (MGS 100 ms) tempi, and correlation (d.f. 78) between ratios and MGS duration for individual participants. Participant MGS 170 ms MGS 100 ms Correlation p B.S. 1.76 2.08.55.001 B.R. 1.45 1.92.76.001 H.R. 1.83 1.72.26.05 R.F. 1.60 1.59.03 n.s. R.B. 1.67 1.81.45.001 S.L. 1.75 1.82.09 n.s. S.K. 1.82 1.65.35.01 V.T. 1.71 1.64.24.05 of metrical interpretation (downbeat location) on the interval ratio. Furthermore, there was no overall difference between synchronization and continuation: Synchronization with a precise rhythm template did not make the produced interval ratios more precise. There was an interaction, however: For 2 3 the mean ratio was larger during synchronization than during continuation, whereas this was not the case for 3 2, F(1,7) 26.8, p.001. The mean ratio was also higher in Session I (1.80) than in Session II (1.72), F(1,7) 7.4, p.03, and was even lower in the continuation condition of Session III (1.65). This could reflect an effect of practice. The decrease between Sessions I and II was larger for 2 3 than for 3 2, F(1,7) 7.2, p.04. Each of the three rhythms in set B yielded two ratios, 3/2a and 3/2b. The mean ratio was 1.73, which indicates substantial enhancement of the contrast between long and short intervals. The mean 3/2a ratio (1.77) was larger than the mean 3/2b ratio (1.69), F(1,7) 9.9, p.02. This implies that the second within-group interval (2b, which preceded the long between-group interval) was longer than the first within-group interval (2a) a form of group-final lengthening (Drake & Palmer, 1993). In contrast to the results for set A, the mean ratio in set B increased significantly from 1.67 to 1.83 as the tempo increased, F(7,49) 5.2, p.0002. However, there were again individual differences, with some participants showing significant increases in both ratios as the tempo increased, others showing a significant increase in the 3/2a ratio but not in the 3/2b ratio, and yet others showing no significant change with tempo of either ratio (see Table 4). The effect of tempo was somewhat larger during synchronization than during continuation, F(7,49) 3.7, p.003. There were no

Production and Synchronization of Uneven Rhythms at Fast Tempi 69 TABLE 4. Mean interval ratios (intercepts of regression line) for set B rhythms at the slowest (MGS 170 ms) and fastest (MGS 100 ms) tempi, and correlation (d.f. 78) between ratios and MGS duration for individual participants. Ptcpt Ratio MGS 170 ms MGS 100 ms Correlation p B.S. 3/2a 1.46 1.80.26.05 3/2b 1.52 1.78.33.01 B.R. 3/2a 1.64 1.80.46.001 3/2b 1.63 1.65.13 n.s. H.R. 3/2a 1.88 1.82.09 n.s. 3/2b 1.74 1.73.03 n.s. R.F. 3/2a 1.76 1.82.34.01 3/2b 1.63 1.71.25.05 R.B. 3/2a 1.40 1.78.64.001 3/2b 1.47 1.72.61.001 S.L. 3/2a 1.34 1.89.65.001 3/2b 1.52 1.75.54.001 S.K. 3/2a 1.77 1.76.02 n.s. 3/2b 1.72 1.79.16 n.s. V.T. 3/2a 1.72 2.01.61.001 3/2b 1.71 1.71.01 n.s. TABLE 5. Mean interval ratios (intercepts of regression line) for set C rhythms at the slowest (MGS 170 ms) and fastest (MGS 100 ms) tempi, and correlation (d.f. 78) between ratios and MGS duration for individual participants. Ptcpt Ratio MGS 170 ms MGS 100 ms Correlation p B.S. 3a/2 1.27 1.80.62.001 3b/2 1.20 2.44.72.001 B.R. 3a/2 1.43 1.74.69.001 3b/2 1.56 1.81.65.001 H.R. 3a/2 1.58 1.69.16 n.s. 3b/2 1.61 1.79.22.05 R.F. 3a/2 1.73 1.78.09 n.s. 3b/2 1.74 1.93.30.01 R.B. 3a/2 1.30 1.70.49.001 3b/2 1.28 1.83.53.001 S.L. 3a/2 1.25 1.62.57.001 3b/2 1.26 1.77.64.001 S.K. 3a/2 1.45 1.72.36.001 3b/2 1.44 1.63.23.05 V.T. 3a/2 1.45 1.55.23.05 3b/2 1.44 1.79.49.001 significant effects involving rhythm (2 2 3 vs. 2 3 2 vs. 3 2 2), which means that metrical interpretation had no consistent effect on timing. An ANOVA conducted on the set B interval ratios in continuation tapping (data from all three sessions) did not show a significant difference between the 3/2a and 3/2b ratios but instead a very reliable Ratio Tempo interaction, F(7,49) 5.6, p.0001: The 3/2a ratio increased more with tempo than did the 3/2b ratio, which means that the relative lengthening of the final within-group interval (2a 2b) emerged as the tempo increased. In addition, the analysis showed that the mean interval ratio in continuation tapping tended to decrease across the three sessions, F(2,14) 4.2, p.04, and that this decrease was more pronounced for the 3/2a ratio than for the 3/2b ratio, F(2,14) 10.0, p.002, especially at a slow tempo, F(7,49) 2.4, p.007. This could reflect again an effect of practice. Each of the rhythms in set C also yielded two ratios, 3a/2 and 3b/2. The mean ratio of 1.64 was somewhat closer to the target value of 1.5 than in the other two rhythm sets. As can be seen in Table 5, only two participants started out with ratios greater than 1.5; most others actually started with ratios smaller than 1.5. The mean 3a/2 ratio (1.59) was smaller than the mean 3b/2 ratio (1.69), F(1,7) 6.1, p.05, which means that 3a 3b. This supports our assumption that 3b functioned as the between-group interval in these rhythms. The ratios decreased from Session I to Session II, F(1,7) 13.9, p.008, but this decrease was smaller for 3a/2 than for 3b/2, F(1,7) 8.9, p.02. Ratios were also smaller in synchronization (1.62) than in continuation (1.67), F(1,7) 12.5, p.01, and a significant triple interaction, F(1,7) 8.1, p.03, suggested that this difference was largest for 3b/2 in Session I. Most importantly, the mean ratio increased significantly with tempo (from 1.48 to 1.88), F(7,49) 9.5, p.0001, but this increase tended to be smaller for 3a/2 than for 3b/2, F(7,49) 3.0, p.02. The effect of tempo on the interval ratios was also significant at the individual level; only two individuals failed to show a significant increase of the 3a/2 ratio (Table 5). As in set B, there were no significant effects involving rhythm (3 3 2 vs. 3 2 3 vs. 2 3 3), which indicates that metrical downbeat location did not play any role. The main results of this section are that (a) the long/short interval ratios were already enhanced at the slowest tempo in rhythm sets A and B; (b) they increased as the tempo increased in sets B and C; and (c) downbeat location within each rhythm set had no influence on interval ratios. INTERVAL VARIABILITY The standard deviations of the inter-tap interval durations provided additional data that were of interest with regard to possible effects of metrical interpretation.

70 B. H. Repp, J. London and P. E. Keller ANOVAs on these data had the variable of interval (2 levels for set A, 3 levels for sets B and C) in lieu of ratio. Three main effects were shown by all three rhythm sets and were significant in each instance: First, as had been fully expected on the basis of earlier interval production studies (e.g., Peters, 1989; Repp, 1997), the long 3 intervals exhibited greater variability than the short 2 intervals. Second, variability decreased as the tempo increased (i.e., as all intervals got shorter). Interestingly, this decrease was especially pronounced in the continuation phase of Session III. Clearly, participants inability to synchronize with rapid subdivisions did not prevent them from producing the rhythms consistently. Third, variability was higher during continuation than during synchronization. In addition, it was noted that variability of both long and short intervals was somewhat higher in set C than in sets A and B. With regard to metrical interpretation (the rhythm variable in the ANOVA), some weak effects were found in sets A and C, but not in set B. In set A, interval variability was somewhat higher for 2 3 than for 3 2, F(1,7) 8.5, p.03, and this difference was larger during continuation than during synchronization, F(1,7) 6.2, p.05. In Session II, the difference was present for both intervals, but in Session I it held only for the long interval, F(1,7) 12.0, p.01. In set C, the Rhythm Interval Tempo interaction approached significance in the main ANOVA, F(28,196) 1.7, p.02, and reached significance in the ANOVA on the continuation data from all three sessions, F(28,196) 1.8, p.01: At the slower tempi, variability of the long intervals was greater in 3 2 3 than in 2 3 3 and 3 3 2, but this difference went away as the tempo increased. Clearly, metrical interpretation had much weaker and less consistent effects on interval variability than did interval duration, tempo, and tapping mode (synchronization vs. continuation). TAPPING FORCE Tapping force (MIDI velocity) was another aspect of the data that was of interest because of potential metrical effects. In particular, our hypothesis was that taps in the same group position might be executed with greater force when they represent metrical downbeats than when they are metrically weak. In addition, we expected to find effects of group position, such that taps following long intervals are more forceful than those following short intervals. In the ANOVAs, group position replaced interval as a variable, and the session variable disappeared because only the MIDI velocities from Session I were analyzed. The effects of temporal grouping and of metrical interpretation are presented graphically in Figure 2. In rhythm set A (Figure 2A), group-initial taps (which followed the long interval) were stronger than group-final taps, as predicted, F(1,6) 21.1, p.004, and this difference increased as the tempo increased, F(7,42) 10.2, p.0001. The difference was also larger in the 2 3 rhythm, where the group-initial tap represented the metrical downbeat, than in the 3 2 rhythm, where FIG 2. Mean key depression velocities for the different rhythms in Session I as a function of rhythmic group position. Db indicates metrical downbeats.

Production and Synchronization of Uneven Rhythms at Fast Tempi 71 the group-final tap represented the downbeat, F(1,6) 22.8, p.004. This means that the metrical accent (i.e., the cognitively imposed downbeat) increased the relative force of the associated tap, as predicted. In rhythm set B (Figure 2B), taps were weaker when they occurred in group-medial position (preceding and following a short interval) than when they were groupinitial or group-final, F(2,12) 16.4, p.0005. The group-initial tap (following a long interval) was weaker than the group-final tap (following a short interval) at slow tempi, but it became stronger than the group-final tap as the tempo increased, F(14,84) 3.9, p.0001, which is the predicted difference. The Rhythm Position interaction was not significant, F(4,24) 1.0, p.42. Thus, there was no effect of metrical interpretation here, only of temporal grouping, although Figure 2B reveals a small tendency for the downbeats to be relatively more forceful. In rhythm set C (Figure 2C), the weakest tap on average was the one following the short interval (considered group-medial), whereas the strongest tap was the one both preceding and following a long interval (considered group-final). These position effects emerged gradually as the tempo increased, F(14,84) 4.0, p.0001. As in rhythm set A, metrical accents made taps relatively more forceful, F(4,24) 4.3, p.01, although this effect was small compared to that of group position. Thus, there were some reliable effects of metrical interpretation here, which is reassuring in that they indicate that participants interpreted the rhythms according to instructions. An unexpected difference that was significant for all three rhythm sets was that participants tapped more strongly during synchronization than during self-paced continuation (Session I). This could represent increasing fatigue in the course of each trial, or greater muscular tension when trying to synchronize with a rhythm template than when tapping in a self-paced manner. Tapping force also decreased substantially as the tempo increased, which may likewise include a fatigue component but probably reflects mainly a reduction in movement amplitude due to the shorter inter-tap intervals. whether asynchronies are reduced when a tap in a given group position represents the metrical downbeat, compared to when it does not. In an ANOVA, this would be reflected in a significant Rhythm Position interaction. The effects of group position and metrical interpretation on the mean asynchronies are shown in Figure 3. In rhythm set A (Figure 3A), the mean asynchrony of the group-initial tap was 9.5 ms, whereas that of the Synchronization ASYNCHRONIES WITH RHYTHM TEMPLATES The fact that interval ratios were not produced accurately implies substantial asynchronies in synchronization with a rhythm template. The asynchronies were necessarily different for taps in different group positions, so a significant main effect of position in an ANOVA is trivial. The main question of interest was FIG 3. Mean asynchronies for the different rhythms in the synchronization conditions of Sessions I and II, as a function of rhythmic group position. Db indicates metrical downbeats.

72 B. H. Repp, J. London and P. E. Keller group-final tap was 15.5 ms, F(1,7) 72.2, p.0001. These values reflect the relative contraction of the short within-group interval and the relative expansion of the long between-group interval, as well as a slight negative asynchrony overall. The Rhythm Position interaction was far from significance, which indicates that metrical interpretation had no reliable effect. No other effects reached significance. In rhythm set B (Figure 3B), the mean asynchronies were 13.1 ms for the group-initial tap, 7.9 ms for the group-medial tap, and 17.3 ms for the group-final tap, F(2,14) 30.5, p.0001. Here there was also a significant main effect of tempo, F(7,49) 6.6, p.0001: Overall, the mean asynchrony became less negative as the tempo increased, changing from 12.0 to 1.4 ms. The Rhythm Position interaction was not significant, F(4,28) 1.9, p.15. Although the Rhythm Position Tempo interaction was reliable, F(28,196) 2.5, p.0001, it was difficult to interpret. In rhythm set C (Figure 3C), the mean asynchrony also became less negative as tempo increased, F(7,49) 14.5, p.0001, changing from 19.0 to 2.3 ms. The main effect of position did not reach significance here, but the Position Tempo interaction did, F(14,98) 5.3, p.0001. The asynchrony of the group-initial tap (preceding the short interval) changed much more with tempo than did the group-medial and group-final asynchronies: from 25.0, 16.2, and 15.9 ms to 22.0, 8.4, and 6.7 ms, respectively. There was also a Session Position interaction, F(2,14) 14.8, p.0005, because the effect of position was larger in Session I than in Session II. The Rhythm Position interaction approached significance, F(4,28) 3.5, p.02, but it did not reflect smaller asynchronies for downbeats. Rather, the asynchronies in the initial and final group positions were more negative when the downbeat fell in these positions. These results offer no support for our hypothesis that downbeats would be associated with smaller asynchronies. Rather, participants strategy seemed to be to minimize the mean asynchrony of each rhythm cycle. VARIABILITY OF ASYNCHRONIES WITH RHYTHM TEMPLATES Variability of asynchronies decreased significantly as a function of increasing tempo in all three rhythm sets. The question of primary interest was whether this variability would be reduced when taps function as metrical downbeats. This would again be reflected in the Rhythm Position interaction. The results are shown in Figure 4. In rhythm set A (Figure 4A), variability was marginally greater for 2 3 than for 3 2, F(1,7) 6.1, p.05. The main effect of position was not significant, nor was the Rhythm Position interaction. In rhythm set B (Figure 4B), variability was greater in the group-initial position than in the medial and final positions, F(2,14) 16.8, p.0002. This may reflect an effect of the long (most variable) interval preceding the group-initial position. However, this difference was present only at the slower tempi; it disappeared as the tempo increased, F(14,98) 4.5, p.0001. Overall, FIG 4. Mean standard deviations of asynchronies for the different rhythms in the synchronization conditions of Sessions I and II, as a function of rhythmic group position. Db indicates metrical downbeats.